Styling autofill dropdown on Search Input

Styling autofill dropdown on Search Input

Hi All, hope this hasn't been discussed before, and it's entirely possible I'm not searching the right thing!
I'm using an input type of "search" which nicely pretties my search field.  It keeps a history of what I type, and when I start typing the same word again it will do an autocomplete-type dropdown (at least in FF and Chrome it does).  I'm guessing it's just the browser doing this, it's not an autocomplete. 

Is there some way to nicely style this dropdown, or possibly remove it altogether if not?

Screenshot attached.

Thanks Much!
Robin